The aim of this paper is to prove that if s &gt; 1 and G is a graph of order n &gt; 4s + 6 satisfying 2 &gt; (4n - 4s - 3) / 3 ; then every matching of G lies on a cycle of length at least n-s and hence, in a path of length at least n - s + 1

GANCARZEWICZ, GRZEGORZ

Portal Czasopism Naukowych (E-Journals)

GRZEGORZ GANCARZEWICZ∗
CYCLES CONTAINING SPECIFIED EDGES IN A
GRAPH
CYKLE ZAWIERAJĄCE WYBRANE KRAWĘDZIE
GRAFU
Abs t r a c t
The aim of this paper is to prove that if s > 1 and G is a graph of order n > 4s+ 6 satisfying
σ2 >
4n− 4s− 3
3
,
then every matching of G lies on a cycle of length at least n− s and hence, in a path of length
at least n− s+ 1.
Keywords: cycle, graph, hamiltonian cycle, hamiltonian path, matching, path
S t r e s z c z e n i e
W pracy udowodniono, że dla s > 1 w dowolnym grafie G rzędu n > 4s+ 6 spełniającym
σ2 >
4n− 4s− 3
3
,
każde skojarzenie jest zawarte w cyklu długości co najmniej n− s i stąd w ścieżce długości co
najmniej n− s+ 1.
Słowa kluczowe: cykl, cykl hamiltonowski, graf, skojarzenie, ścieżka, ścieżka hamiltonowska
∗Institute of Mathematics, Cracow University of Technology, Poland; ggancarzewicz@gmail.com
38
1. Introduction
We consider only finite graphs without loops and multiple edges. By V or V(G) we
denote the vertex set of graph G and respectively, by E or E(G), the edge set of G.
By dG(x) or d(x) we denote the degree of a vertex x in the graph G.
In the proof we will only use oriented cycles and paths. Let C be a cycle and
x ∈ V (C) , then x− is the predecessor of x and x+ is its successor.
Let us introduce the σk.
Definition 1.1. Let G be a graph and k > 0.
σk = min{
k∑
i=1
d(xi) : {x1, . . . , xk} ⊂ V(G) and independent }
In 1960, O. Ore [5] proved the following:
Theorem 1.1. Let G be a graph on n > 3 vertices. If G satisfies
σ2 > n
then G is hamiltonian.
The condition for degree sum in Theorem 1.1 is called an Ore condition or a Ore
type condition for graph G.
The Ore condition for a graph G :
σ2 > l
can also be written as:
If x, y ∈ V(G), xy 6∈ E(G), then: d(x) + d(y) > l.
There is also a similar condition, proved by V. Chvátal in [3], under which graph
G has a hamiltonian path.
Theorem 1.2. If G is a graph on n > 3 vertices satisfying
σ2 > n− 1, (1.1)
then G has a hamiltonian path.
We shall call a set of k independent edges of graph G a k-matching or simply a
matching.
About graphs with every k-matching in a hamiltonian cycle or path Las Vergnas
obtained the following two results:
39
Theorem 1.3. Let G be a graph on n > 3 vertices and let k be an integer such that
0 6 k 6 n2 . If G satisfies
σ2 > n+ k − 1 ,
then every k-matching of G lies in a hamiltonian path.
Theorem 1.4. Let G be a graph on n > 3 vertices and let k be an integer such that
0 6 k 6 n2 . If G satisfies
σ2 > n+ k ,
then every k-matching of G lies in a hamiltonian cycle.
K.A. Berman proved in [1] the following result conjectured by R. Häggkvist in [4].
Theorem 1.5. Let G be graph of order n. If G satisfies
σ2 > n+ 1,
then every matching lies in a cycle.
Now we shall define a family of graphs Gn. If n+23 is an integer, Gn is a family of
graphs:
G =
n+ 2
3
K1 ? H,
where ? denotes join and H is a graph of order 2n−23 containing a perfect matching.
Otherwise, Gn is empty.
In 1983 Wojda [6] proved the following Ore type theorem:
Theorem 1.6. Let G be a graph on n > 3 vertices.If G satisfies
σ2 >
4n− 4
3
.
Then every matching of G lies in a hamiltonian cycle or G ∈ Gn.
In this paper, we shall find an Ore type condition under which every matching in a
graph G lies in a cycle of length at least n− s and hence, in a path of length at least
n− s+ 1.
For the notation and terminology not defined above, a good reference should be [2].
40
2.Results
We proved the following improvement of Theorem 1.6 for matchings.
Theorem 2.1. Let s > 1 and let G be a graph of order n > 4s+ 6 satisfying
σ2 >
4n− 4s− 3
3
. (2.1)
Then, every matching of G lies on a cycle of length at least n− s and hence in a path
of length at least n− s+ 1.
The special case of this theorem for s = 1 is:
Corollary 2.2. Let G be a graph of order n > 10, satisfying
σ2 >
4n− 7
3
.
Then, every matching of G lies on a cycle of length at least n − 1 and hence in a
hamiltonian path.
Obviously for k > n−23 , the bound for σ2 is lower in Corollary 2.2 than the bound
from Theorem 1.3.
Suppose that s > 1 is such that n > 4s+ 6 and n+2s3 > 2 is an integer.
Now consider the graph G′ = (n+2s3 − 1)K1 ∗K 2n−2s3 , where ∗ denotes the join of
graphs.
We shall define a graph G′′ as a graph obtained from G′ by adding an external vertex
x adjacent only to 2n−2s3 − 1 vertices from K 2n−2s3 i.e. we take V(G
′′) = V(G′) ∪ {x},
next we choose an arbitrary vertex h0 ∈ V(K 2n−2s
3
) and we put E(G′′) = E(G′)∪{xh :
h ∈ V(K 2n−2s
3
) \ {h0}}. Note theat G′′ is a graph of order n.
Let u ∈ V(K1), then dG′′(u) = 2n−2s3 − 1 and dG′′(x) + dG′′(h0) = 4n−4s−33 , the
graph G′′ satisfies the assumptions of Theorem 2.1, but violates those of Theorem 1.6.
So, Theorem 1.6 and Theorem 2.1 are independent.
It is easy to check that even Corollary 2.2 cannot be obtained as a corollary of
Theorem 1.6 by adding to the graph G an external vertex x adjacent to all vertices
and removing an edge from the hamiltonian cycle in G ∪ {x}. In this case, G ∪ {x}
does not satisfy the assumptions of Theorem 1.6.
Obviously, for k > n−23 , the bound for σ2 is lower in Theorem 2.2 than the bound
from Theorem 1.3.
41
3. Proof
Proof of Theorem 2.1:
Take any matching M of G. Without loss of generality we can assume that M is
maximal, i.e. for any matching M ′ of G if M ⊂M ′, then M = M ′.
Observe that since n > 4s+ 6, we have 4n−4s−33 > n+ 1. If the graph G satisfies
the assumptions of Theorem 2.1 then it also satisfies the assumptions of Theorem 1.5.
From Theorem 1.5, we know that there is a cycle containing M.
Consider a cycle C containing M of maximal length. If |V(C)| > n− s the proof is
finished. We suppose now that |V(C)| 6 n− s− 1 and we give an arbitrary orientation
to C.
Since M is maximal, the set V(G \ C) is independent.
Since s > 1 we have |V(G \ C)| > 2 and therefore we have two vertices x and
y ∈ V(G \ C) such that xy 6∈ E(G) and from (2.1) we have:
d(x) + d(y) > 4n− 4s− 3
3
. (3.1)
Note that since V(G \ C) is independent we have:
dG\C(x) = dG\C(y) = 0. (3.2)
On cycle C, consider a family of paths Qi, i ∈ {1, . . . , k}, obtained from C by the
removal of the edges of matching M.
Note that:
k∑
i=1
|V(Qi)| = |V(C)|. (3.3)
Since M is maximal, |V(Qi)| = 2 or |V(Qi)| = 3. (3.4)
Remark 3.1. If w ∈ V(G \C), then w cannot be adjacent to two consecutive vertices
on any path Qi, for i ∈ {1, . . . , k}.
Suppose that w ∈ V(G \C) and we have a vertex u ∈ V(Qi) such that u+ ∈ V(Qi)
and wu, wu+ ∈ E. In this case, the cycle:
C ′ : uwu+ . . . u−u
contains M and is longer than C, contradiction with the choice of C.
42
Case 1: |V(Qi)| = 2
From Remark 3.1, we know that dQi(x) 6 1 and dQi(y) 6 1 and so:
dQi(x) + dQi(y) 6 2 = |V(Qi)|. (3.5)
Case 2: |V(Qi)| = 3
In this case, from Remark 3.1 we know that dQi(x) 6 2 and dQi(y) 6 2. Note that
if Qi : qi1qi2qi3 it is possible that x and y are adjacent at the same time to qi1 and qi2.
From the above we have:
dQi(x) + dQi(y) 6 4 = |V(Qi)|+ 1. (3.6)
Consider now the set I = {Qi : |V(Qi)| = 3 and Qi : qi1qi2qi3}, l = |I|. Observe
that since M is maximal we have:
1. The set VI = {qi2 : Qi ∈ I} is independent and |VI | = l.
2. If dQi(x) = 2, then xqi2 6∈ E and if dQi(y) = 2, then yqi2 6∈ E.
We have l + 2 independent vertices in G : VI ∪ {x, y}.
Thus:
dC(x) + dC(y) =
k∑
i=1
dQi(x) + dQi(y)
6
k∑
i=1
|V(Qi)|+ l = |V(C)|+ l. (3.7)
From (3.2) and (3.7), we have:
d(x) + d(y) =
k∑
i=1
dQi(x) + dQi(y) 6 |V(C)|+ l. (3.8)
Since |V(C)| 6 n− s− 1 and M is maximal, we have l 6 n−s−13 and so:
d(x) + d(y) 6 |V(C)|+ l 6 n− s− 1 + n− s− 1
3
=
4n− 4s− 4
3
. (3.9)
Since d(x) + d(y) is an integer, from (3.9), we have:
d(x) + d(y) 6
⌊4n− 4s− 4
3
⌋
<
4n− 4s− 3
3
, (3.10)
43
a contradiction with (3.1) and the proof is finished.

Re f e r e n c e s
[1] Berman K.A., Proof of a conjecture of Haggkvist on cycles and independent edges,
Discrete Mathematics 46, 1983, 9—13.
[2] Bondy J.A. and Murty U.S.R., Graph theory with applications, The Macmillan
Press LTD, London 1976.
[3] Chvátal V., On Hamilton’s ideals, J. Combin. Theory B 12, 1972, 163—168.
[4] Häggkvist R., On F-hamiltonian graphs in Graph Theory and Related Topics, ed.
J.A. Bondy and U.S.R. Murty, Academic Press N.Y. 1979 219—231.
[5] Ore O., Note on hamiltonian circuits, Amer. Math. Monthly 67, 1960, 55.
[6] Wojda, A.P. , Hamiltonian cycles through matchings, Demonstratio Mathematica
XXI 2, 1983, 547—553.



CYCLES CONTAINING SPECIFIED EDGES IN A GRAPH

http://www.ejournals.eu/sj/index.php/Cz/article/download/1611/1620

CYCLES CONTAINING SPECIFIED EDGES IN A GRAPH

Abstract

Similar works

Full text

Available Versions

Portal Czasopism Naukowych (E-Journals)